p-group, metabelian, nilpotent (class 3), monomial
Aliases: M4(2)⋊11D4, C42.377C23, (C2×C8)⋊18D4, C8.4(C2×D4), C8⋊3D4⋊6C2, C8⋊4D4⋊18C2, (C4×C8)⋊28C22, C4⋊C4.241D4, C2.22(D4○D8), (C22×D8)⋊19C2, (C2×D8)⋊21C22, C4⋊1D4⋊7C22, C22⋊C4.81D4, C4.13(C22×D4), C8⋊C4⋊49C22, C8.12D4⋊16C2, C4.42(C4⋊1D4), (C2×C4).353C24, (C2×C8).560C23, (C2×Q16)⋊45C22, C23.455(C2×D4), C4.4D4⋊9C22, C8○2M4(2)⋊13C2, (C2×SD16)⋊17C22, (C2×D4).119C23, (C2×Q8).107C23, C22.29C24⋊13C2, C22.19(C4⋊1D4), (C22×C8).272C22, C22.613(C22×D4), (C22×C4).1043C23, (C22×D4).379C22, C42⋊C2.326C22, (C2×M4(2)).273C22, (C2×C4○D8)⋊22C2, (C2×C8⋊C22)⋊25C2, (C2×C4).139(C2×D4), C2.32(C2×C4⋊1D4), (C2×C4○D4).159C22, SmallGroup(128,1887)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 716 in 298 conjugacy classes, 108 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×28], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×6], D4 [×28], Q8 [×4], C23, C23 [×14], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×6], M4(2) [×4], D8 [×18], SD16 [×12], Q16 [×2], C22×C4, C22×C4 [×2], C2×D4 [×6], C2×D4 [×20], C2×Q8 [×2], C4○D4 [×8], C24 [×2], C4×C8 [×2], C8⋊C4 [×2], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×4], C4⋊1D4 [×4], C22×C8, C2×M4(2), C2×D8, C2×D8 [×8], C2×D8 [×4], C2×SD16 [×6], C2×Q16, C4○D8 [×4], C8⋊C22 [×8], C22×D4 [×2], C2×C4○D4 [×2], C8○2M4(2), C8⋊4D4 [×2], C8.12D4 [×2], C8⋊3D4 [×4], C22.29C24 [×2], C22×D8, C2×C4○D8, C2×C8⋊C22 [×2], M4(2)⋊11D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C4⋊1D4 [×4], C22×D4 [×3], C2×C4⋊1D4, D4○D8 [×2], M4(2)⋊11D4
Generators and relations
G = < a,b,c,d | a8=b2=c4=d2=1, bab=cac-1=a5, dad=a-1, cbc-1=dbd=a4b, dcd=c-1 >
►On 32 points
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)
(2 6)(4 8)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)
(1 13 29 19)(2 10 30 24)(3 15 31 21)(4 12 32 18)(5 9 25 23)(6 14 26 20)(7 11 27 17)(8 16 28 22)
(1 28)(2 27)(3 26)(4 25)(5 32)(6 31)(7 30)(8 29)(9 12)(10 11)(13 16)(14 15)(17 24)(18 23)(19 22)(20 21)
G:=sub<Sym(32)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32), (1,13,29,19)(2,10,30,24)(3,15,31,21)(4,12,32,18)(5,9,25,23)(6,14,26,20)(7,11,27,17)(8,16,28,22), (1,28)(2,27)(3,26)(4,25)(5,32)(6,31)(7,30)(8,29)(9,12)(10,11)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21)>;
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(26,30)(28,32), (1,13,29,19)(2,10,30,24)(3,15,31,21)(4,12,32,18)(5,9,25,23)(6,14,26,20)(7,11,27,17)(8,16,28,22), (1,28)(2,27)(3,26)(4,25)(5,32)(6,31)(7,30)(8,29)(9,12)(10,11)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21) );
G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23),(26,30),(28,32)], [(1,13,29,19),(2,10,30,24),(3,15,31,21),(4,12,32,18),(5,9,25,23),(6,14,26,20),(7,11,27,17),(8,16,28,22)], [(1,28),(2,27),(3,26),(4,25),(5,32),(6,31),(7,30),(8,29),(9,12),(10,11),(13,16),(14,15),(17,24),(18,23),(19,22),(20,21)])
Matrix representation ►G ⊆ GL6(𝔽17)
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 16 | 2 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 14 |
| 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 14 | 3 | 0 | 0 |
G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,14,14,0,0,0,0,3,14,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,16,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,14,14,0,0,0,0,14,3,0,0,14,14,0,0,0,0,14,3,0,0] >;
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 8A | 8B | 8C | 8D | 8E | ··· | 8J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D4○D8 |
| kernel | M4(2)⋊11D4 | C8○2M4(2) | C8⋊4D4 | C8.12D4 | C8⋊3D4 | C22.29C24 | C22×D8 | C2×C4○D8 | C2×C8⋊C22 | C22⋊C4 | C4⋊C4 | C2×C8 | M4(2) | C2 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
M_{4(2)}\rtimes_{11}D_4 % in TeX
G:=Group("M4(2):11D4"); // GroupNames label
G:=SmallGroup(128,1887);
// by ID
G=gap.SmallGroup(128,1887);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,184,521,2804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^5,d*a*d=a^-1,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations